Simplify. Express your answer as a single fraction in simplest form. \[ \frac{z}{z+7}+\frac{z}{z^{2}+5 z-14} \]

To simplify the expression \[ \frac{z}{z+7} + \frac{z}{z^2 + 5z - 14}, \] we first need to factor the quadratic in the second fraction, \( z^2 + 5z - 14 \). We look for two numbers that multiply to \(-14\) and add up to \(5\). The numbers \(7\) and \(-2\) fit this requirement. Thus, we can factor: \[ z^2 + 5z - 14 = (z + 7)(z - 2). \] Now, we can rewrite the original expression using this factorization: \[ \frac{z}{z+7} + \frac{z}{(z+7)(z-2)}. \] Next, we need a common denominator to combine the fractions. The common denominator is \((z + 7)(z - 2)\). We will rewrite the first fraction: \[ \frac{z}{z+7} = \frac{z(z - 2)}{(z + 7)(z - 2)}. \] Now we can combine the two fractions: \[ \frac{z(z - 2)}{(z + 7)(z - 2)} + \frac{z}{(z + 7)(z - 2)} = \frac{z(z - 2) + z}{(z + 7)(z - 2)}. \] This simplifies the numerator: \[ z(z - 2) + z = z^2 - 2z + z = z^2 - z. \] Thus, we have: \[ \frac{z^2 - z}{(z + 7)(z - 2)}. \] Now, we can factor the numerator \(z^2 - z\): \[ z^2 - z = z(z - 1). \] So, the expression can be rewritten as: \[ \frac{z(z - 1)}{(z + 7)(z - 2)}. \] Finally, we conclude that the simplified expression is: \[ \boxed{\frac{z(z - 1)}{(z + 7)(z - 2)}}. \]